In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

where is an open set and and denote the gradient and the Laplacian of u with respect to the space variables. They prove that for such initial data there exist two pairs of initial data for which the solution is global, and such that

and the nonlinearity and external force term g will be specified. The main results are focused on the relationships among the growth exponent p of the nonlinearity and well-posedness. They show that (i) even if p is up to the supercritical range,

that is, , the well-posedness and the longtime behavior of the so-

lutions of the equation are of the characters of the parabolic equation; (ii) when

where is a bounded domain in with the smooth boundary, , and are nonlinear functions, and is an external force term. They prove that in strictly positive stiffness factors and supercritical nonlinearity case, there exists a global finite-dimensional attractor in the natural energy space endowed with strong topology.

For the most of the scholars represented by Yang Zhijian have studied all kinds of low order Kirchhoff equations and only a small number of scholars have studied the blow-up and asymptotic behavior of solutions for higher-order Kirchhoff equation. So, in this context, we study the high-order Kirchhoff equation is very meaningful. In order to study the high-order nonlinear Kirchhoff equation with the damping term, we borrow some of Li Yan’s [5] partial assumptions (2.1) - (2.3) for the nonlinear term g in the equation. In order to prove that the lemma 1, we have improved the results from assumptions (2.1) - (2.3) such that. Then, under all assumptions, we prove

that the equation has a unique smooth solution

and obtain the solution semigroup has global attractor. Finally, we prove the equation has finite Hausdorff dimensions and Fractal dimensions by reference to the literature [7] .

For more related results we refer the reader to [6] [7] [8] [9] [10] . In order to make these equations more normal, in section 2 and in section 3, some assumptions, notations and the main results are stated. Under these assumptions, we prove the existence and uniqueness of solution, then we obtain the global attractors for the problems (1.1) - (1.3). According to [6] [7] [8] [9] [10] , in section 4, we consider that the global attractor of the above mentioned problems (1.1) - (1.3) has finite Hausdorff dimensions and fractal dimensions.

2. Preliminaries

For convenience, we denote the norm and scalar product in by and;

, , , , ,

According to [5] , we present some assumptions and notations needed in the proof of our results. For this reason, we assume nonlinear term satisfies that

(H1) Setting then

(2.1)

(H2) If

(2.2)

where

(H3) There exist constant, such that

(2.3)

(H4) There exist constant, such that

(2.4)

(2.5)

where;

For every, by (H1)-(H3) and apply Poincaré inequality, there exist constants, such that

Proof. Under the conditions of Theorem 3.1, it exists the solution semigroup S(t), , here.

(1) From Lemma 1 to Lemma 2, we can get that is a bounded set that includes in the ball,

(3.20)

This shows that is uniformly bounded in.

(2) Furthermore, for any, when, we have

(3.21)

So we get is the bounded absorbing set.

(3) Since is compact embedded, which means that the bounded set in is the compact set in, so the semigroup operator S(t) exists a compact global attractor.

4. The Estimates of the Upper Bounds of Hausdorff and Fractal Dimensions for the Global Attractor

We rewrite the problems (1.1) - (1.3):

(4.1)

(4.2)

(4.3)

Let, where is a bounded domain in with smooth boundary, q is positive constant, and m is positive integer. The linearized equations of the above equations as follows:

(4.4)

(4.5)

Let, is the solution of problems (4.4) - (4.5). We can prove that the problems (4.4) - (4.5) have a unique solution The equation (4.4) is the linearized equation by the Equation (4.17). Define the

mapping, here, let,

, let, , ,

,.

Lemma 4.1 [6] Assume H is a Hilbert space, is a compact set of H. is a continuous mapping, satisfy the follow conditions.

1);

2) If is Fréchet differentiable, it exists is a bounded linear differential operator, that is

The proof of lemma 4.1 see ref. [6] is omitted here. According to Lemma 4.1, we can get the following theorem :

Theorem 4.1. [6] [7] Let is the global attractor that we obtain in section 3.In that case, has finite Hausdorff dimensions and Fractal dimensions in

,that is.

Let, let, is an isomorphic mapping. So let is the global attractor of, then is also the global attractor of, and they have the same dimensions. Then satisfies as follows:

(4.6)

(4.7)

where

(4.8)

(4.9)

(4.10)

(4.11)

where. The initial condition (4.5) can be written in the following form:

(4.12)

We take, then consider the corresponding n solutions: of the initial values: in the Equations (4.10) - (4.11). So there is

. from

, we get , here u is the solution of problems (4.1)-(4.3); represents the outer product, Tr reprsents the trace, is an orthogonal projection from the space to the subspace spanned by.

For a given time, let. is the

standard orthogonal basis of the space.

From the above, we have

(4.13)

where is the inner product in.Then; .

(4.14)

where

Now, suppose that, according to theorem 3.3, is a bounded absorbing set in..

Then there is a to make the mapping. At the same time, there are the following results:

(4.15)

where meets:. Comprehensive above can be obtained:

(4.16)

, due to is a standard orthogonal basis in. So

(4.17)

Almost to all t, making

(4.18)

So

(4.19)

Let us assume that, is equivalent to Then

(4.20)

According to (4.19), (4.20), so

(4.21)

Therefore, the Lyapunov exponent of (or) is uniformly bounded.

(4.22)

From what has been discussed above, it exists, a and r are constants, then

(4.23)

(4.24)

(4.25)

(4.26)

According to the reference [6] [7] , we immediately to the Hausdorff dimension and fractal dimension are respectively.

5. Conclusion

In this paper, we prove that the higher-order nonlinear Kirchhoff equation with linear damping in has a unique smooth solution. Fur- ther, we obtain the solution semigroup has global attractor. Finally, we prove the equation has finite Hausdorff dimensions and Fractal dimensions in.

Acknowledgements

The authors express their sincere thanks to the aonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.

Fund

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.